One weight inequality for Bergman projection and Calderón operator induced by radial weight
Authors: Martín Reyes Francisco , Ortega Pedro , Peláez José , Rättyä Jouni 
Published: September 2021

Let

ω
\omega

and

ν
\nu

be radial weights on the unit disc of the complex plane such that

ω
\omega

admits the doubling property
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript 0 less-than-or-equal-to r greater-than 1 Endscripts StartFraction integral Subscript r Superscript 1 Baseline omega left-parenthesis s right-parenthesis d s Over integral Subscript StartFraction 1 plus r Over 2 EndFraction Superscript 1 Baseline omega left-parenthesis s right-parenthesis d s EndFraction greater-than normal infinity">

sup

0
≤
r
>
1

∫
r
1

ω
(
s
)

d
s

∫

1
+
r

2

1

ω
(
s
)

d
s

>
∞

\sup _{0\le r>1}\frac {\int _r^1 \omega (s)\,ds}{\int _{\frac {1+r}{2}}^1 \omega (s)\,ds}>\infty

. Consider the one weight inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis reverse-solidus dag right-parenthesis EndLabel double-vertical-bar upper P Subscript omega Baseline left-parenthesis f right-parenthesis double-vertical-bar Subscript upper L Sub Subscript nu Sub Superscript p Subscript Baseline less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript upper L Sub Subscript nu Sub Superscript p Subscript Baseline comma 1 greater-than p greater-than normal infinity comma EndLayout">

(\dag)

‖

P
ω

(
f
)

‖

L
ν
p

≤
C
‖
f

‖

L
ν
p

,

1
>
p
>
∞
,

\begin{equation} \|P_\omega (f)\|_{L^p_\nu }\le C\|f\|_{L^p_\nu },\quad 1>p>\infty ,\tag {\dag } \end{equation}

for the Bergman projection

P
ω

P_\omega

induced by

ω
\omega

. It is shown that the Muckenhoupt-type condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript p Baseline left-parenthesis omega comma nu right-parenthesis equals sup Underscript 0 less-than-or-equal-to r greater-than 1 Endscripts StartFraction left-parenthesis integral Subscript r Superscript 1 Baseline s nu left-parenthesis s right-parenthesis d s right-parenthesis Superscript StartFraction 1 Over p EndFraction Baseline left-parenthesis integral Subscript r Superscript 1 Baseline s left-parenthesis StartFraction omega left-parenthesis s right-parenthesis Over nu left-parenthesis s right-parenthesis Superscript StartFraction 1 Over p EndFraction Baseline EndFraction right-parenthesis Superscript p prime Baseline d s right-parenthesis Superscript StartFraction 1 Over p prime EndFraction Baseline Over integral Subscript r Superscript 1 Baseline s omega left-parenthesis s right-parenthesis d s EndFraction greater-than normal infinity">

A
p

(
ω
,
ν
)
=

sup

0
≤
r
>
1

(

∫
r
1

s
ν
(
s
)

d
s
)

1
p

(

∫
r
1

s

(

ω
(
s
)

ν
(
s

)

1
p

)

p
′

d
s
)

1

p
′

∫
r
1

s
ω
(
s
)

d
s

>
∞

\begin{equation*} A_p(\omega ,\nu )=\sup _{0\le r>1}\frac {\left (\int _r^1 s\nu (s)\,ds \right )^{\frac {1}{p}}\left (\int _r^1 s\left (\frac {\omega (s)}{\nu (s)^{\frac 1p}}\right )^{p’}\,ds \right )^{\frac {1}{p’}}}{\int _r^1 s\omega (s)\,ds}>\infty \end{equation*}

is necessary for (

\dag ) to hold, and sufficient if

ν
\nu

is of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row nu left-parenthesis s right-parenthesis equals omega left-parenthesis s right-parenthesis left-parenthesis integral Subscript r Superscript 1 Baseline s omega left-parenthesis s right-parenthesis d s right-parenthesis Superscript alpha EndLayout">

ν
(
s
)
=
ω
(
s
)

(

∫
r
1

s
ω
(
s
)

d
s
)

α

\begin{align*} \nu (s)=\omega (s)\left (\int _r^1 s\omega (s)\,ds \right )^\alpha \end{align*}

for some

−
1
>
α
>
∞

-1>\alpha >\infty

. This result extends the classical theorem due to Forelli and Rudin for a much larger class of weights. In addition, it is shown that for any pair

(
ω
,
ν
)

(\omega ,\nu )

of radial weights the Calderón operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript omega Superscript star Baseline left-parenthesis f right-parenthesis left-parenthesis z right-parenthesis plus upper H Subscript omega Baseline left-parenthesis f right-parenthesis left-parenthesis z right-parenthesis equals integral Subscript 0 Superscript StartAbsoluteValue z EndAbsoluteValue Baseline f left-parenthesis s StartFraction z Over StartAbsoluteValue z EndAbsoluteValue EndFraction right-parenthesis StartFraction s omega left-parenthesis s right-parenthesis d s Over integral Subscript s Superscript 1 Baseline t omega left-parenthesis t right-parenthesis d t EndFraction plus StartFraction integral Subscript StartAbsoluteValue z EndAbsoluteValue Superscript 1 Baseline f left-parenthesis s StartFraction z Over StartAbsoluteValue z EndAbsoluteValue EndFraction right-parenthesis s omega left-parenthesis s right-parenthesis d s Over integral Subscript StartAbsoluteValue z EndAbsoluteValue Superscript 1 Baseline s omega left-parenthesis s right-parenthesis d s EndFraction">

H
ω
⋆

(
f
)
(
z
)
+

H
ω

(
f
)
(
z
)
=

∫

0

|

z

|

f

(
s

z

|

z

|

)

s
ω
(
s
)

d
s

∫
s
1

t
ω
(
t
)

d
t

+

∫

|

z

|

1

f

(
s

z

|

z

|

)

s
ω
(
s
)

d
s

∫

|

z

|

1

s
ω
(
s
)

d
s

\begin{equation*} H^\star _\omega (f)(z)+H_\omega (f)(z) =\int _{0}^{|z|} f\left (s\frac {z}{|z|}\right )\frac {s\omega (s)\,ds}{\int _s^1 t\omega (t)\,dt} +\frac {\int _{|z|}^1f\left (s\frac {z}{|z|}\right ) s\omega (s)\,ds}{\int _{|z|}^1 s\omega (s)\,ds} \end{equation*}

is bounded on

L
ν
p

L^p_\nu

if and only if

A
p

(
ω
,
ν
)
>
∞

A_p(\omega ,\nu )>\infty

.