bilinear restriction estimates for surfaces with curvatures of different signs recently the sharp l 2 bilinear adjoint restriction estimates for the cone and the paraboloid were established by wolff and tao respectively their results rely on the fact that for the cone and the paraboloid the nonzero principal curvatures have the same sign we generalize those bilinear restriction estimates to surfaces with curvatures of different signs

restriction theorems for a surface with negative curvature we prove a bilinear restriction theorem for a surface of negative curvature this is the analogue of the results of t wolff 19 and t tao 14 15 for cones and paraboloids as a consequence we obtain an almost sharp linear restriction theorem

thresholding in a generalized model for translation invariant systems abstract in a previous paper we introduced a generalized model for translation invariant ti operators we considered the space of all maps from an abelian group g to called lg fuzzy sets where is a complete lattice ordered group and we defined ti operators on this space in this paper thresholding is considered in the same general framework and in this regard positive ti operators are studied the threshold of an lg fuzzy set a at level t is defined to be the set g g a g t also a ti operator is defined to be positive if its kernel consists of elements which are positive except possibly at some points which are as the main result of this note it is proved that a positive and isotone ti operator commutes with thresholding if and only if it has a crisp kernel or base also the theorem is used to derive a similar result for an operator whose kernel is bounded from below

hermitian geometry and complex space time we consider a complex hermitian manifold of complex dimensions four with a hermitian metric and a chern connection it is shown that the action that determines the dynamics of the metric is unique provided that the linearized einstein action coupled to an antisymmetric tensor is obtained in the limit when the imaginary coordinates vanish the unique action is of the chern simons type when expressed in terms of the kahler form the antisymmetric tensor field has gauge transformations coming from diffeomorphism invariance in the complex directions the equations of motion must be supplemented by boundary conditions imposed on the hermitian metric to give in the limit of vanishing imaginary coordinates the low energy effective action for a curved metric coupled to an antisymmetric tensor

a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

restriction estimates for some surfaces with vanishing curvatures abstract we obtain bilinear restriction estimates for surfaces with vanishing curvatures as application we also prove new linear restriction estimates for some class of conic surfaces

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

reconstruction in a generalized model for translation invariant systems abstract we consider translation invariant ti operators on the set of maps from an abelian group g to called lg fuzzy sets where 0 is a complete lattice ordered group by defining minkowski and morphological operations on and considering order preserving operators we prove a reconstruction theorem this theorem which is called the strong reconstruction theorem srt is similar to the convolution theorem in the theory of linear and shift invariant systems and states that for an order preserving ti operator y one can explicitly compute y a for any a from a specific subset of called the base of y the introduced framework is a general model for the theory of translation invariant systems and srt shows the consistency of it for the special cases when g r z srt implies the results of maragos and schafer

generalized strichartz estimates and scattering for 3d zakharov system we obtain scattering for the 3d zakharov system with non radial small data in the energy space with angular regularity of degree one the main ingredient is a generalized strichartz estimate for the schrodinger equation in the space of l2 angular integrability

improved restriction estimate for hyperbolic surfaces in r3 abstract recently l guth improved the restriction estimate for the surfaces with strictly positive gaussian curvature in r 3 in this paper we extend his restriction estimate to the quadratic surfaces with strictly negative gaussian curvature in r 3

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

matrix weighted besov spaces nazarov treil and volberg defined matrix ap weights and extended the theory of weighted norm inequalities on l p to the case of vector valued functions we develop some aspects of littlewood paley function space theory in the matrix weight setting in particular we introduce matrix weighted homogeneous besov spaces b q p w and matrix weighted sequence besov spaces b q p w as well as b q p a q where the a q are reducing operators for w under any of three different conditions on the weight w we prove the norm equivalences f b q p w s q q b q p w s q q b q p a q where s q q is the vector valued sequence of transform coefficients of f in the process we note and use an alternate more explicit characterization of the matrix ap class furthermore we introduce a weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on b q p w if w is doubling we also obtain the boundedness of almost diagonal operators on b q p w under any of the three conditions on w this leads to the boundedness of convolution and non convolution type calderon zygmund operators czos on b q p w in particular the hilbert transform we apply these results to wavelets to show that the above norm equivalence holds if the transform coefficients are replaced by the wavelet coefficients finally we construct inhomogeneous matrix weighted besov spaces b q p w and show that results corresponding to those above are true also for the inhomogeneous case

fourier restriction to a hyperbolic cone abstract using a bilinear restriction theorem of lee and a bilinear to linear argument of stovall we obtain the conjectured range of fourier restriction estimates for a conical hypersurface in r 4 with hyperbolic cross sections

morphological representation of discrete and binary images a general theory for the morphological representation of discrete and binary images is presented the basis of this theory relies on the generation of a set of nonoverlapping segments of an image via repeated erosions and set transformations which in turn produces a decomposition that guarantees exact reconstruction the relationship between the proposed representation and some existing shape analysis tools e g discrete size transform pattern spectrum skeletons is investigated thus introducing the representation as the basis of a unified theory for geometrical image analysis particular cases of the general representation scheme are shown to yield a number of useful image decompositions which are directly related to various forms of morphological skeletons the relationship between the representation and the various forms of morphological skeletons is studied as a result of this study a unified theory is developed for the mathematical description of the morphological skeleton decomposition of discrete and binary images

the optimal trilinear restriction estimate for a class of hypersurfaces with curvature abstract in 4 nearly optimal l 1 trilinear restriction estimates in r n 1 are established under transversality assumptions only in this paper we show that the curvature improves the range of exponents by establishing l p estimates for any p 2 n 4 3 n 2 in the case of double conic surfaces the exponent 2 n 4 3 n 2 is shown to be the universal threshold for the trilinear estimate

endpoint bilinear restriction theorems for the cone and some sharp null form estimates recently wolff 25 obtained a nearly sharp l 2 bilinear restriction theorem for bounded subsets of the cone in general dimension we obtain the endpoint of wolff s estimate and generalize to the case when one of the subsets is large as a consequence we are able to deduce some nearly sharp l p null form estimates

morphological filters part i their set theoretic analysis and relations to linear shift invariant filters this paper examines the set theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations which involve supremum infimum operations and additions binary signals are classified as sets and multilevel signals as functions two set theoretic representations of signals are reviewed filters are classified as set processing sp or function processing fp conditions are provided for certain fp filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as sp filters the basic morphological operations of set erosion dilation opening and closing are related to minkowski set operations and are used to construct fp morphological filters emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions the latter case allows the definition of morphological convolutions and correlations toward this goal various properties of fp morphological filters are also examined linear shift invariant filters due to their translation invariance are uniquely characterized by their kernel which is a special collection of input signals increasing linear filters are represented as the supremum of erosions by their kernel functions if the filters are also discrete and have a finite extent impulse response they can be represented as the supremum of erosions only by their minimal with respect to a signal ordering kernel functions stable linear filters can be represented as the sum of at most two weighted suprema of erosions these results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal shaping strategies

a sharp l p l q fourier restriction theorem for a conical surface of finite type in this paper we study the fourier restriction problem for certain conical surfaces where the sections are curves of finite type we obtain the sharp l p l q range for this surfaces our methods rely on a variation of the affine arclength measure

sharp null form estimates for the wave equation null form estimates from dot h alpha 1 times dot h alpha 2 to l q t l r x for the wave equation in bbb r n 1 are studied for n ge4 we obtain the sharp null form estimates except for the endpoints for n 2 3 we obtain the estimates under the additional assumption 4 n q when 2 r

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

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

nonlinear profile decomposition for the h 12 h 12 rd energy subcritical wave equation abstract we consider the nonlinear wave equation u t t u u u 4 d 1 0 in dimensions d 2 this equation is critical in h 1 2 h 1 2 r d and subcritical with respect to the energy we prove the nonlinear profile decomposition the proof must address the lack of compactness due to the lorentz symmetry

conjecture and improved extension theorems for paraboloids in the finite field setting we study the extension estimates for paraboloids in d dimensional vector spaces over finite fields mathbb f q with q elements we use the connection between l 2 based restriction estimates and l p rightarrow l r extension estimates for paraboloids as a consequence we improve the l 2 rightarrow l r extension results obtained by lewko and lewko proc am math soc 140 2013 2028 2012 in even dimensions d ge 6 and odd dimensions d 4 ell 3 for ell in mathbb n our results extend the consequences for 3 d paraboloids due to lewko adv math 270 1 457 479 2015 to higher dimensions we also clarifies conjectures on finite field extension problems for paraboloids

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

strichartz estimates via the schrodinger maximal operator we consider the schrodinger operator e it delta acting on initial data f in dot h s we show that an affirmative answer to a question of carleson concerning the sharp range of s for which lim t to 0 e it delta f x f x a e x in mathbb r n would imply an affirmative answer to a question of planchon concerning the sharp range of q and r for which e it delta is bounded in l x q mathbb r n l r t mathbb r when n 2 we unconditionally improve the range for which the mixed norm estimates hold

weighted restriction estimates using polynomial partitioning we use the polynomial partitioning method of guth j amer math soc 29 2016 371 413 to prove weighted fourier restriction estimates in r3 with exponents p that range between 3 and 3 25 depending on the weight as a corollary to our main theorem we obtain new non weighted local and global restriction estimates for compact c surfaces s r3 with strictly positive second fundamental form for example we establish the global restriction estimate ef lp r3 f lq s in the full conjectured range of exponents for p 3 25 up to the sharp line and the global restriction estimate ef lp f l2 s for p 3 and certain sets r3 of infinite lebesgue measure as a corollary to our main theorem we also obtain new results on the decay of spherical means of fourier transforms of positive compactly supported measures on r3 with finite dimensional energies

improved bound for the bilinear bochner riesz operator we study l p times l q rightarrow l r bounds for the bilinear bochner riesz operator mathcal b alpha alpha 0 in mathbb r d d ge 2 which is defined by open image in new window we make use of a decomposition which relates the estimates for mathcal b alpha to the square function estimates for the classical bochner riesz operators in consequence we significantly improve the previously known bounds

weak version of restriction estimates for spheres and paraboloids in finite fields abstract we study l p l r restriction estimates for algebraic varieties in d dimensional vector spaces over finite fields unlike the euclidean case if the dimension d is even then it is conjectured that the l 2 d 2 d 3 l 2 stein tomas restriction result can be improved to the l 2 d 4 d 4 l 2 estimate for both spheres and paraboloids in finite fields in this paper we show that the conjectured l p l 2 restriction estimate holds in the specific case when test functions under consideration are restricted to d coordinate functions or homogeneous functions of degree zero to deduce our result we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in d 1 dimensions

a refinement of the strichartz inequality on the saddle and applications abstract we consider the nonlinear nonelliptic schrodinger equation defined by i t u x 2 y 2 u u 2 u 0 with initial datum in l 2 r 2 we show that if the solution blows up in finite time then there is a mass concentration phenomenon near the blow up time the key ingredient is a refinement of the strichartz inequality on the saddle

restriction and kakeya phenomena for finite fields the restriction and kakeya problems in euclidean space have received much attention in the last few decades and they are related to many problems in harmonic analysis partial differential equations pdes and number theory in this paper we initiate the study of these problems on finite fields in many cases the euclidean arguments carry over easily to the finite setting and are in fact somewhat cleaner but there are some new phenomena in the finite case which deserve closer study

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

the schrodinger equation along curves and the quantum harmonic oscillator abstract we consider the schrodinger equation associated to the harmonic oscillator i t u h u where h x 2 with initial data in the sobolev space h s r d with d 2 and s 3 8 we prove almost everywhere convergence of the solution to its initial data as time tends to zero which improves a result of yajima 1990 30 to this end we consider almost everywhere convergence for the free schrodinger along curves as it turns out these problems are more or less equivalent to that of the free schodinger equation along vertical lines our results are obtained by showing the equivalence of related maximal estimates

linear adjoint restriction estimates for paraboloid we prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices this result generalizes the paraboloid restriction estimate in radial case from shao rev mat iberoam 25 2009 1127 1168 as well as the result from miao et al proc ams 140 2012 2091 2102 as an application we show a local smoothing estimate for a solution of the linear schrodinger equation under the assumption that the initial datum has additional angular regularity

on the restriction of the fourier transform to a conical surface let r be the su rface of a circular cone in r3 we show that if 1 p 4 3 1 q 3 1 1 p and f e lp r3 then the fourier transform of f belongs to lq r da for a certain natural measure a on r following p tomas we also establish bounds for restrictions of fourier transforms to conic annuli at the endpoint p 4 3 with logarithmic growth of the bound as the thickness of the annulus tends to zero

fractal strichartz estimate for the wave equation abstract we consider strichartz estimates for the wave equation with respect to general measures which satisfy certain growth conditions in r 3 1 we obtain the sharp estimate and in higher dimensions improve the previous results

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

mixed norm strichartz type estimates for hypersurfaces in three dimensions in their work ikromov and muller fourier restriction for hypersurfaces in three dimensions and newton polyhedra princeton university press princeton 2016 proved the full range l p l 2 fourier restriction estimates for a very general class of hypersurfaces in mathbb r 3 which includes the class of real analytic hypersurfaces in this article we partly extend their results to the mixed norm case where the coordinates are split in two directions one tangential and the other normal to the surface at a fixed given point in particular we resolve completely the adapted case and partly the non adapted case in the non adapted case the case when the linear height h text lin phi is below two is settled completely

morphological filters part ii their relations to median order statistic and stack filters this paper extends the theory of median order statistic os and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions dilations openings closings and open closings that commute with thresholding the max min representation of os filters is introduced by showing that any median or other os filter is equal to a maximum of erosions moving local minima and also to a minimum of dilations moving local maxima thus os filters can be computed by a closed formula that involves a max min on prespecified sets of numbers and no sorting stack filters are established as the class of filters that are composed exactly of a finite number of max min operations the kernels of median os and stack filters are collections of input signals that uniquely represent these filters due to their translation invariance the max min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal with respect to a signal ordering kernel elements and also to a minimum of dilations by minimal kernel elements of dual filters the representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum of products expressions of boolean functions it is also shown that median filtering and its iterations of any signal by convex 1 d windows is bounded below by openings and above by closings a signal is a root fixed point of the median iff it is a root of both an opening and a closing the open closing and clos opening yield median roots in one pass suppress impulse noise similarly to the median can discriminate between positive and negative noise impulses and are computationally less complex than the median some similar results are obtained for 2 d median filtering

estimates for the x ray transform restricted to line complexes of kirillov type let there be given a piecewise continuous rectifiable curve r r n let g 1 n m 1 n be the usual grassmannian bundle in r n define an n dimensional submanifold m r n of m 1 n as the set of all copies of g 1 n along the curve following krillov we know that a nice function f x can be recovered from its x ray transform r 1 n f on m r n if and only if the curve intersects almost every affine hyperplane

complexified gravity in noncommutative spaces the presence of a constant background antisymmetric tensor for open strings or d branes forces the space time coordinates to be noncommutative this effect is equivalent to replacing ordinary products in the effective theory by the deformed star product an immediate consequence of this is that all fields get complexified the only possible noncommutative yang mills theory is the one with u n gauge symmetry by applying this idea to gravity one discovers that the metric becomes complex we show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry u 1 d 1 instead of the usual so 1 d 1 the final theory depends on a hermitian tensor containing both the symmetric metric and antisymmetric tensor in contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant the results are then generalized to noncommutative spaces

an endline bilinear cone restriction estimate for mixed norms we prove an l 2 times l 2 rightarrow l t ql x p bilinear fourier extension estimate for the cone when p q are on the critical line 1 q frac n 1 2 1 1 p this extends previous results by wolff tao and lee vargas

duality in a generalized model for translation invariant systems abstract in a previous paper we introduced a generalized model for translation invariant ti operators in this model we considered the space of all maps from an abelian group g to u called lg fuzzy sets where is a complete lattice ordered group and we defined ti operators on this space also in that paper we proved strong reconstruction theorem to show the consistency of this model this theorem states that for an order preserving ti operator y one can explicitly compute y a for any a from a specific subset of called the base of y in this paper duality is considered in the same general framework and in this regard continuous ti operators are studied this kind of operators are characterized in terms of critical points of their kernels a critical point is defined to be a point at which there exists a net which converges to when is the value of a nontrivial limit function of the kernel at that point a critical point is called to be of first type if for every such net there exists some other point at which this net converges to and it is called to be of second type otherwise as the main result of this paper we prove that a welldefined isotone ti operator is continuous if and only if it has no critical point of second type moreover in this case a ti operator has a continuous extension by duality

residuated semigroups and morphological aspects of translation invariant systems abstract the main goal of this paper is to verify classical properties of morphological operators within the general model of translation invariant ti systems in this model ti operators are defined on the space of lg fuzzy sets i e a g in which g is an abelian group and is a complete lattice ordered group a ti operator y is an operator on which is invariant under translation on g and as groups we consider the generalization of minkowski addition d on and we emphasize that is an involutive residuated topological monoid we verify all properties of traditional set theoretic morphological operators as well as classical representations matheron 1967 for openings closures and granulometries in this general setting we also study spectral skeleton decompositions in this model using the same techniques as in the crisp case goutsias and schonfeld 1991 this formal approach not only clarifies the role of morphological operators but also it gives rise to simpler and clearer proofs using standard results of the theory of residuated semigroups and categories

a generalized fuzzy mathematical morphology and its application in robust 2 d and 3 d object representation in this paper the generalized fuzzy mathematical morphology gfmm is proposed based on a novel definition of the fuzzy inclusion indicator fii fii is a fuzzy set used as a measure of the inclusion of a fuzzy set into another that is proposed to be a fuzzy set it is proven that the fii obeys a set of axioms which are proposed to be extensions of the known axioms that any inclusion indicator should obey and which correspond to the desirable properties of any mathematical morphology operation the gfmm provides a very powerful and flexible tool for morphological operations the binary and grayscale mathematical morphologies can be considered as special cases of the proposed gfmm an application for robust skeletonization and shape decomposition of two dimensional 2 d and three dimensional 3 d objects is presented simulation examples show that the object reconstruction from their skeletal subsets that can be achieved by using the gfmm is better than by using the binary mathematical morphology in most cases furthermore the use of the gfmm for skeletonization and shape decomposition preserves the shape and the location of the skeletal subsets and spines

multi scale bilinear restriction estimates for general phases we prove adjoint bilinear restriction estimates for general phases at different scales in the full non endpoint mixed norm range and give bounds with a sharp and explicit dependence on the phases these estimates have applications to high low frequency interactions for solutions to partial differential equations as well as to the linear restriction problem for surfaces with degenerate curvature as a consequence we obtain new bilinear restriction estimates for elliptic phases and wave klein gordon interactions in the full bilinear range and give a refined strichartz inequality for the klein gordon equation in addition we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves

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

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

radial multipliers and restriction to surfaces of the fourier transform in mixed norm spaces in this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces l rad p l ang 2 left mathbb r n right we produce sharp bounds for the restriction of the fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier we also present some results for the discrete restriction conjecture and state an intriguing open problem

a sobolev estimate for the adjoint restriction operator in this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption we obtain the optimal h s l q estimates and their mixed norm generalizations as applications we prove some weighted strichartz estimates for the propagator varphi rightarrow e it delta alpha 2 varphi alpha 0