global existence for energy critical waves in 3 d domains we prove that the defocusing quintic wave equation with dirichlet boundary conditions is globally well posed on for any smooth compact domain the main ingredient in the proof is an spectral projector estimate obtained recently by smith and sogge combined with a precise study of the boundary value problem

on kakeya nikodym type maximal inequalities we show that for any dimension d 3 one can obtain wolff s l d 2 2 bound on kakeya nikodym maximal function in r d for d 3 without the induction on scales argument the key ingredient is to reduce to a 2 dimensional l 2 estimate with an auxiliary maximal function we also prove that the same l d 2 2 bound holds for nikodym maximal function for any manifold m d g with constant curvature which generalizes sogge s results for d 3 to any d 3 as in the 3 dimensional case we can handle manifolds of constant curvature due to the fact that in this case two intersecting geodesics uniquely determine a 2 dimensional totally geodesic submanifold which allows the use of the auxiliary maximal function

estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis we study the boundedness problem for maximal operators mathcal m associated with averages along smooth hypersurfaces s of finite type in 3 dimensional euclidean space for p 2 we prove that if no affine tangent plane to s passes through the origin and s is analytic then the associated maximal operator is bounded on l p left mathbb r 3 right if and only if p h s where h s denotes the so called height of the surface s defined in terms of certain newton diagrams for non analytic s we obtain the same statement with the exception of the exponent p h s our notion of height h s is closely related to a n varchenko s notion of height h for functions such that s can be locally represented as the graph of after a rotation of coordinates

riesz means associated with certain product type convex domain in this paper we study the maximal operators t and the convolution operators t associated with multipliers of the form 1 max 0 1 n 2 0 1 n 2 r2 rn 2 n 3 we prove that t satisfies the sharp weak type p p inequality on hp rn when 2 n 1 2n 1 n 1p 1 12 we also obtain that t is bounded from lp rn to lp rn for max 2 1p 12 12 0 and 1 p the indicated ranges of parameters p and cannot be improved

radial fourier multipliers of lp r2 abstract we give sufficient conditions in terms of differentiability and growth properties for a radial function to be an lp r2 fourier multiplier these conditions are in the nature of best possible

on the lp norm of spectral clusters for compact manifolds with boundary we use microlocal and paradifferential techniques to obtain l8 norm bounds for spectral clusters associated with elliptic second order operators on two dimensional manifolds with boundary the result leads to optimal lq bounds in the range 2 q for l2 normalized spectral clusters on bounded domains in the plane and more generally for two dimensional compact manifolds with boundary we also establish new sharp lq estimates in higher dimensions for a range of exponents qn q

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

bilinear kakeya nikodym averages of eigenfunctions on compact riemannian surfaces abstract we obtain an improvement of the bilinear estimates of burq gerard and tzvetkov 6 in the spirit of the refined kakeya nikodym estimates 2 of blair and the second author we do this by using microlocal techniques and a bilinear version of hormander s oscillatory integral theorem in 7

refined and microlocal kakeya nikodym bounds of eigenfunctions in higher dimensions we prove a kakeya nikodym bound on eigenfunctions and quasimodes which sharpens a result of the authors blair and sogge in anal pde 8 747 764 2015 and extends it to higher dimensions as in the prior work the key intermediate step is to prove a microlocal version of these estimates which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic geodesic flow in a companion paper blair and sogge in j differ geom 2015 it will be seen that these sharpened estimates yield improved l q m bounds on eigenfunctions in the presence of nonpositive curvature when 2 q frac 2 d 1 d 1

strichartz estimates in the frame of modulation spaces abstract in this paper we study the strichartz estimates for three types of differential equations in the framework of modulation spaces the equations considered here include the wave equation the schrodinger equation and the nonelliptic schrodinger equation inhomogeneous equations of the above three types are also studied

a few endpoint geodesic restriction estimates for eigenfunctions we prove a couple of new endpoint geodesic restriction estimates for eigenfunctions in the case of general 3 dimensional compact manifolds after a tt argument simply by using the l 2 boundedness of the hilbert transform on mathbb r we are able to improve the corresponding l 2 restriction bounds of burq gerard and tzvetkov duke math j 138 445 486 2007 and hu forum math 6 1021 1052 2009 also in the case of 2 dimensional compact manifolds with nonpositive curvature we obtain improved l 4 estimates for restrictions to geodesics which by holder s inequality and interpolation implies improved l p bounds for all exponents p 2 we do this by using oscillatory integral theorems of hormander ark mat 11 1 11 1973 greenleaf and seeger j reine angew math 455 35 56 1994 and phong and stein int math res notices 4 49 60 1991 along with a simple geometric lemma lemma 3 2 about properties of the mixed hessian of the riemannian distance function restricted to pairs of geodesics in riemannian surfaces we are also able to get further improvements beyond our new results in three dimensions under the assumption of constant nonpositive curvature by exploiting the fact that in this case there are many totally geodesic submanifolds

global existence for energy critical waves in 3 d domains neumann boundary conditions we prove that the defocusing quintic wave equation with neumann boundary conditions is globally well posed on h 1 l 2 for any smooth compact domain r 3 the proof relies on one hand on l p estimates for the spectral projector and on the other hand on a precise analysis of the boundary value problem which turns out to be much more delicate than in the case of dirichlet boundary conditions

local smoothing for operators failing the cinematic curvature condition in this paper we examine a class of averaging operators which exhibit local smoothing that is viewed as a function of space and time variables the operators yield more smoothing than the fixed time estimates sogge showed in a more general setting that if these operators satisfy a cinematic curvature condition they will exhibit some local smoothing c d sogge propagation of singularities and maximal functions in the plane invent math 104 1991 231 251 here we translate this condition into the setting of averaging operators in the plane we prove that cinematic curvature is not necessary for local smoothing to occur exhibiting a class of operators which fail the cinematic curvature condition but still satisfy a local smoothing estimate furthermore the amount of local smoothing exhibited by these operators is strictly less than that conjectured for operators satisfying the cinematic curvature condition

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

a maximal function for families of hilbert transforms along homogeneous curves let h u be the hilbert transform along the parabola t ut 2 where u in mathbb r for a set u of positive numbers consider the maximal function mathcal h u f sup h u f u in u we obtain an essentially optimal result for the l p operator norm of mathcal h u when 2 p infty the results are proved for families of hilbert transforms along more general nonflat homogeneous curves

on the cone multiplier in r3 abstract we prove the sharp l 3 bounds for the cone multiplier in r 3 and the associated square function which is known as mockenhaupt s square function

strichartz and localized energy estimates for the wave equation in strictly concave domains we prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous dirichlet or neumann conditions are imposed by restricting the solution to small frequency dependent space time collars of the boundary it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates mixed norm estimates of strichartz and square function type follow as a result using the energy estimates to control error terms which arise in a wave packet parametrix construction while the latter estimates are not new for dirichlet conditions the present approach provides an avenue for treating these estimates when neumann conditions are imposed the method also treats schr o dinger equations with time independent coefficients

eigenvalue bounds for dirac and fractional schrodinger operators with complex potentials abstract we prove lieb thirring type bounds for fractional schrodinger operators and dirac operators with complex valued potentials the main new ingredient is a resolvent bound in schatten spaces for the unperturbed operator in the spirit of frank and sabin 11

boundaries of a convex set and interpolation sets the structure of the boundary of a convex set c is a field of intensive research in functional analysis the classical boundary is the set ext c of the extreme points of c at least when some compactness is assumed and c is recovered from ext c by means of the integral representation theory in this paper a more general notion of boundary definition 1 i is considered such a boundary needs not contain or even meet the extreme points and thus the classical tools are not available however through r c james s technique and a remarquable result of s simons a convex set can often be recovered in a strong sense from its boundary sect 1 tight connections are established between this notion lacunarity in harmonic analysis sect 2 banach spaces containing or not n sect 3 compactness and duality theory sect 4 finally sect 5 is devoted to miscellaneous results and questions in the spirit of the work and to examples showing the necessity of the assumptions we made as a general remark we did not systematically try to state the results in their most general form as far as it was not necessary to introduce new ideas for doing so this is specially true for the results of sect 2 where we worked in the duality 2 of the little fourier analysis although the techniques we used remain valid within the general frame of locally compact abelian groups

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

global parametrices and dispersive estimates for variable coefficient wave equations in this article we consider variable coefficient time dependent wave equations in mathbb r times mathbb r n using phase space methods we construct outgoing parametrices and prove strichartz type estimates globally in time this is done in the context of c 2 metrics which satisfy a weak asymptotic flatness condition at infinity

maximal operators associated to families of flat curves in the plane bourgain showed that if c has non vanishing curvature the inequality 1 1 holds for p 2 see b in this paper we shall consider a situation when the curvature is allowed to vanish of finite order on a finite set of isolated points we shall need the following definition definition1 1 let c i r where i is a compact interval in r and c is smooth we say that c is finite type if c x c x0 does not vanish of infinite order for any x0 i and any unit vector we shall also need a more precise definition which would specify the order of vanishing at each point let a0 denote a point in the compact interval i we can always find a smooth function such that in a small neighborhood of a0 c s s s where s i definition1 2 let c be defined as before let c s s s in a small neighborhood of a0 we say that c is finite type m at a0 if a0 0 for 1 k m and a0 6 0 our main result is the following theorem 1 1 let c be a finite type curve which is finite type m at a0 let mtf x

weak type estimates for maximal operators with a cylindric distance function we establish sharp weak type estimates for the maximal operators t associated with cylindric riesz means for functions on h p 3 when 4 5 3 p 5 2

propagation of singularities for the wave equation on conic manifolds for the wave equation associated to the laplacian on a compact manifold with boundary with a conic metric with respect to which the boundary is metrically a point the propagation of singularities through the boundary is analyzed under appropriate regularity assumptions the diffracted non direct wave produced by the boundary is shown to have sobolev regularity greater than the incoming wave

estimates on fractional power dissipative equations in function spaces abstract we study the fractional power dissipative equations whose fundamental semigroup is given by e t with 0 by using an argument of duality and interpolation we extend space time estimates of the fractional power dissipative equations in lebesgue spaces to the hardy spaces and the modulation spaces these results are substantial extensions of some known results as applications we study both local and global well posedness of the cauchy problem for the nonlinear fractional power dissipative equation u t u u m u for initial data in the modulation spaces

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

categoricity properties for computable algebraic fields we examine categoricity issues for computable algebraic fields such fields behave nicely for computable dimension we show that they cannot have finite computable dimension greater than 1 however they behave less nicely with regard to relative computable categoricity we give a structural criterion for relative computable categoricity of these fields and use it to construct a field that is computably categorical but not relatively computably categorical finally we show that computable categoricity for this class of fields is 0 complete

boundaries of compact convex sets and fragmentability if x is a compact convex set in a real locally convex space b x is said to be its boundary if every affine continuous function on x attains its maximum at some point of b we study relations between fragmentability of b and the whole set x as a byproduct we obtain a characterization of separable asplund spaces we also study the possibility of finding the haar system in a boundary of a metrizable compact convex set

strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation the aim of this article is threefold first we use the fbi transform to set up a calculus for partial differential operators with nonsmooth coefficients next this calculus allows us to prove strichartz type estimates for the wave equation with nonsmooth coefficients finally we use these strichartz estimates to improve the local theory for second order nonlinear hyperbolic equations 1 introduction the fbi transform is in a way similar to the complex fourier transform in that for each function in rn it provides a representation as a holomorphic function in r2n however in the case of the fbi transform we can identify naturally r2n with the phase space t rn for a pseudodifferential operator with smooth symbol acting on functions in rn one can produce by conjugation a corresponding formal series acting on func tions in r2n9 for which the first term is exactly the multiplication by the symbol this series converges and has a nice representation in the weyl calculus provided that the symbol of the operator is analytic this is how the fbi transform has been used in the study of partial differential operators with analytic coefficients see 12 13 where this machinery is developed here in a way we do the opposite we look at operators with nonsmooth coefficients approximate the conjugated operator by a partial sum of the formal series and prove error estimates in the simplest case the approximate conjugate operator is exactly the multiplication by the symbol this is also related to the cordoba fefferman wave packet transform in 3 in the third section we use the error estimates to reduce the strichartz estimates for second order hyperbolic operators with nonsmooth coefficients to weighted lp lq l2 estimates for the fbi transform these in turn can be proved in the usual fashion using appropriate oscillatory integral esti mates in the last part of the article we explain how one can use the new strichartz type estimates to improve the local theory for nonlinear hyperbolic equations beyond the classical setup these results are not sharp and will be improved in subsequent articles

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

counterexamples to strichartz estimates for the wave equation in domains we consider the wave equation inside a strictly convex domain of dimension 2 and provide counterexamples to optimal strichartz estimates such estimates inside convex domains lose regularity when compared to the flat case at least for a subset of the usual range of indices mainly due to microlocal phenomena such as caustics which are generated in arbitrarily small time near the boundary

weighted l p estimates for rough bi parameter fourier integral operators abstract fourier integral operators play an important role in applications to partial differential equations the main purpose of this paper is to investigate the weighted l p boundedness of the bi parameter fourier integral operators or fios for short of the following form t f x r n 1 r n 2 e i x a x f d d where for x x 1 x 2 r n 1 r n 2 r n 1 0 and r n 2 0 the amplitude function a x l b s m and the phase function is of the form x 1 x 1 2 x 2 when 1 l 2 r n 1 r n 1 0 2 l 2 r n 2 r n 2 0 satisfy certain differential conditions we will establish the weighted l w p boundedness of t under the assumption that the weight w is in the class of the product muckenhoupt a p r n 1 r n 2 weights

concerning nikodym type sets in 3 dimensional curved spaces in this paper we shall give some natural partial extensions to the curved space setting of results of bourgain 1 and wolff 21 concerning lower bounds for the dimension of compliments of nikodym sets in euclidean space recall that a classical nikodym set is a subset of 1 1 x 1 1 of lebesgue measure one which has the property that for each x e f there is a line yx so that x nf x because of this the relative compliment q 1 1 x 1 1 f must be a set of measure zero with the property that if q is the set of points x for which there is a line segment y through x with iq n axf a then iq ji 0 for every 0 7 3 if iq i 0 here as before tyx denotes a line segment through x of length ae 0 this lower bound was later improved by wolff 21 to dim q 5 2 if iq i o in both works the lower bounds on the dimension were obtained for somewhat more general sets lower bounds for analogous sets in higher dimensions were also obtained in 1 and 21 the strongest to date are those of wolff 21 showing that if q c ri and iq i 0 for some ae 0 then dim q n 2 2 let us now consider extensions of this result to the curved 3 dimensional setting to this end we shall let m3 g denote a paracompact 3 dimensional riemannian manifold with metric ds2 zgjk x dxjdxk given x e m3 we let ya denote the set of all geodesics containing x with arclength ae that is i y a abusing the classical terminology somewhat we now define nikodym type subsets of m3

strichartz estimates for second order hyperbolic operators with nonsmooth coefficients ii in an earlier work we have proved some estimates of strichartz type for second order hyperbolic operators with c s coefficients for 0 s 1 here we improve these estimates and extend the range of s to 0 s 2 in particular for s 2 we obtain the full strichartz estimates

linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces abstract we obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces as applications we improve the known estimates for oscillatory integrals

bochner riesz means with respect to a rough distance function the generalized bochner riesz operator s r may be defined as s r f x f 1 1 r f x where p is an appropriate distance function and f 1 is the inverse fourier transform the behavior of s r on l p r d x r is described for p d 1 max d 1 a rough distance function we conjecture that this operator is bounded on r d x r when max d 1 2 1 p 1 2 0 and p 2 6 d 3 and unbounded when p 2 6 d 3 this conjecture is verified for large ranges of p

spectral cluster estimates for metrics of sobolev regularity we investigate spectral cluster estimates for compact manifolds equipped with a riemannian metric whose regularity is determined by its inclusion in a sobolev space of sufficiently high order the problem is reduced to obtaining l p estimates for the wave equation which are shown by employing wave packet techniques

weights of boundaries of compact convex sets let b be a boundary of a compact convex set x we prove that the weight of x equals the weight of b provided b is lindelof or x is a standard compact convex set and b is the set of extreme points of x

computable categoricity for algebraic fields with splitting algorithms a computably presented algebraic field f has a splitting algorithm if it is decidable which polynomials in f x are irreducible there we prove that such a field is computably categorical iff it is decidable which pairs of elements of f belong to the same orbit under automorphisms we also show that this criterion is equivalent to the relative computable categoricity of f

strichartz type estimates and the well posedness of an energy critical 2d wave equation in a bounded domain abstract we study the well posedness of the cauchy problem with dirichlet or neumann boundary conditions associated to an h 1 critical semilinear wave equation on a smooth bounded domain r 2 first we prove an appropriate strichartz type estimate using the l q spectral projector estimates of the laplace operator our proof follows burq lebeau and planchon 2008 4 then we show the global well posedness when the energy is below or at the threshold given by the sharp moser trudinger inequality finally in the supercritical case we prove an instability result using the finite speed of propagation and a quantitative study of the associated ode with oscillatory data

spectral cluster estimates for c 1 1 metrics in this paper we establish l p norm bounds for spectral clusters on compact manifolds under the assumption that the metric is c 1 1 precisely we show that the l p estimates proven by sogge in the case of smooth metrics hold under this limited regularity assumption it is known by examples of smith sogge that such estimates fail for c 1 metrics if 1

on global existence for nonlinear wave equations outside of convex obstacles the authors prove global existence of small solutions to a semilinear wave equation outside of convex obstacles this extends results of christodoulou and klainerman who handled the minkowski space version the proof is a compromise of the methods of christodoulou and klainerman it relies on local estimates proved earlier by smith and sogge together with classical energy decay estimates for the wave equation of morawetz lax and phillips

topologies weaker than the weak topology of a banach space abstract in this paper we investigate some properties of the compact subsets of banach spaces x endowed with topologies of the kind x b where b is a norming subset of the dual unit ball b x assuming that b x is sequentially compact we prove that the krein smulian theorem holds for norm bounded x b compact subsets of x and we state that the convex x b compact subsets of x have the weak radon nikod m property when b x is sequentially compact and x has either the separable complementation property or x is weakly lindelof for instance when b x is corson compact we prove that the x b compact subsets resp x b compact convex subsets of x are fragmented by the norm of x resp have the radon nikod m property so if b x is a corson compact then the compact subsets of the space x x b are radon nikod m compact and thus sequentially compact we apply the previous results to prove that if b x is sequentially compact and b is assumed to be a boundary of b x then the norm bounded x b compact subsets of x are weakly compact which partially answers a problem posed by g godefroy we give among others applications to spaces of vector valued bochner integrable functions as well as to spaces of countably additive measures

estimates for a class of oscillatory integrals and decay rates for wave type equations this paper investigates higher order wave type equations of the form formula see text where the symbol formula see text is a real non degenerate elliptic polynomial of the order formula see text on formula see text using methods from harmonic analysis we first establish global pointwise time space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the cauchy problem of such wave equations these estimates are then used to establish pointwise in time formula see text estimates on the wave solution in terms of the initial conditions

strichartz estimates and maximal operators for the wave equation in r3 abstract we prove sharp strichartz type estimates in three dimensions including some which hold in reverse space time norms for the wave equation with potential these results are also tied to maximal operator estimates studied by rogers villarroya of which we prove a sharper version as a sample application we use these results to prove the local well posedness and the global well posedness for small initial data of semilinear wave equations in r 3 with quintic or higher monomial nonlinearities

logarithmic improvements in l p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature we consider the problem of proving l p bounds for eigenfunctions of the laplacian in the high frequency limit in the presence of nonpositive curvature and more generally manifolds without conjugate points in particular we prove estimates at the critical exponent p c frac 2 d 1 d 1 where a spectrum of scenarios for phase space concentration must be ruled out our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical l p bounds of the second author

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

boundaries for banach spaces determine weak compactness a boundary for a real banach space is a subset of the dual unit sphere with the property that each element of the banach space attains its norm on an element of that subset trivially the pointwise convergence with respect to such a boundary is coarser than the weak topology on the banach space the boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets

l p regularity of averages over curves and bounds for associated maximal operators we prove that for a finite type curve in r 3 the maximal operator generated by dilations is bounded on if for sufficiently large p we also show the endpoint lp l p i p regularity result for the averaging operators for large p the proofs make use of a deep result of thomas wolff about decompositions of cone multipliers

improved bounds for stein s square functions supported in part by nrf grant 2009 0072531 korea micinn grant mtm2010 16518 spain erc grant 277778 europe and nsf grant 0652890 usa

strichartz type estimates for fractional heat equations abstract we obtain strichartz estimates for the fractional heat equations by using both the abstract strichartz estimates of keel tao and the hardy littlewood sobolev inequality we also prove an endpoint homogeneous strichartz estimate via replacing l x r n by bmo x r n and a parabolic homogeneous strichartz estimate meanwhile we generalize the strichartz estimates by replacing the lebesgue spaces with either besov spaces or sobolev spaces moreover we establish the strichartz estimates for the fractional heat equations with a time dependent potential of an appropriate integrability as an application we prove the global existence and uniqueness of regular solutions in spatial variables for the generalized navier stokes system with l r r n data

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

sharp resolvent estimates outside of the uniform boundedness range in this paper we are concerned with resolvent estimates for the laplacian delta in euclidean spaces uniform resolvent estimates for delta were shown by kenig et al duke math j 55 2 329 347 1987 who established rather a complete description of the lebesgue spaces allowing such estimates however the problem of obtaining sharp l p l q bounds depending on z has not been considered in a general framework which admits all possible p q in this paper we present a complete picture of sharp l p l q resolvent estimates which may depend on z we also obtain the sharp resolvent estimates for the fractional laplacians and a new result for the bochner riesz operators of negative index