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Let $0<\alpha<1$ and define

$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$

The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of $Hf(x):=\int \frac{f(y)}{|x-y|^{\alpha}}dy$ which majorizes $|Tf(x)|$ by the triangle inequality, when $f>0$. It asserts that $$ \|Tf\|_{q}\leq C \|f\|_{p}\qquad (1)$$ where $C$ is a constant independent of $f$, if and only if $$\frac{1}{q}-\frac{1}{p}=1-\alpha\qquad (2)$$ The necessity of (2) for (1) can be easily deduced by scaling.

My question:

Is (2) necessary for the inequality $$ \|Tf\|_{q}\leq C_{1} \|f\|_{p}\qquad (3)\qquad ?$$

It seems that $T$ does not have any scaling properties. I cannot come up with counterexamples that show that (3) is false outside the range of exponents described by (2). Is the boundedness of $T$ known ?

Let $0<\alpha<1$ and define

$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$

The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of $Hf(x):=\int \frac{f(y)}{|x-y|^{\alpha}}dy$ which majorizes $|Tf(x)|$ by the triangle inequality, when $f>0$. It asserts that $$ \|Tf\|_{q}\leq C \|f\|_{p}\qquad (1)$$ where $C$ is a constant independent of $f$, if and only if $$1<p,q<\infty\quad\frac{1}{p}-\frac{1}{q}=1-\alpha\qquad (2)$$ The necessity of (2) for (1) can be easily deduced by scaling.

My question:

Is (2) necessary for the inequality $$ \|Tf\|_{q}\leq C_{1} \|f\|_{p}\qquad (3)\qquad ?$$

It seems that $T$ does not have any scaling properties. I cannot come up with counterexamples that show that (3) is false outside the range of exponents described by (2). Is the boundedness of $T$ known ?

Let $0<\alpha<1$ and define $$ Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy. $$ The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of  $Hf(x):=\int \frac{f(y)}{|x-y|^{\alpha}}dy$ which majorizes $|Tf(x)|$ by the triangle inequality, when $f>0$. It asserts that $$ \|Hf\|_{q}\leq C \|f\|_{p}\label{1}\tag{1} $$ where $C$ is a constant independent of $f$, if and only if $$\frac{1}{q}-\frac{1}{p}=1-\alpha\label{2}\tag{2}$$ The necessity of \eqref{2} for \eqref{1} can be easily deduced by scaling.

My question:

Is \eqref{2} necessary for the inequality $$ \|Tf\|_{q}\leq C_{1} \|f\|_{p}\label{3}\tag{3}\;?$$

It seems that $T$ does not have any scaling properties. I cannot come up with counterexamples that show that \eqref{3} is false outside the range of exponents described by \eqref{2}. Is the boundedness of $T$ known ?

Let $0<\alpha<1$ and define

$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$

The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of  $Hf(x):=\int \frac{f(y)}{|x-y|^{\alpha}}dy$ which majorizes $|Tf(x)|$ by the triangle inequality, when $f>0$. It asserts that $$ \|Tf\|_{q}\leq C \|f\|_{p}\qquad (1)$$ where $C$ is a constant independent of $f$, if and only if $$\frac{1}{q}-\frac{1}{p}=1-\alpha\qquad (2)$$ The necessity of (2) for (1) can be easily deduced by scaling.

My question:

Is (2) necessary for the inequality $$ \|Tf\|_{q}\leq C_{1} \|f\|_{p}\qquad (3)\qquad ?$$

It seems that $T$ does not have any scaling properties. I cannot come up with counterexamples that show that (3) is false outside the range of exponents described by (2). Is the boundedness of $T$ known ?

Let $0<\alpha<1$ and define

$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$

The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of  $Hf(x):=\int \frac{f(y)}{|x-y|^{\alpha}}dy$ which majorizes $|Tf(x)|$ by the triangle inequality, when $f>0$. It asserts that $$ \|Hf\|_{q}\leq C \|f\|_{p}\qquad (1)$$ where $C$ is a constant independent of $f$, if and only if $$\frac{1}{q}-\frac{1}{p}=1-\alpha\qquad (2)$$ The necessity of (2) for (1) can be easily deduced by scaling.

My question:

Is (2) necessary for the inequality $$ \|Tf\|_{q}\leq C_{1} \|f\|_{p}\qquad (3)\qquad ?$$

It seems that $T$ does not have any scaling properties. I cannot come up with counterexamples that show that (3) is false outside the range of exponents described by (2). Is the boundedness of $T$ known ?

Let $0<\alpha<1$ and define $$ Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy. $$ The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of  $Hf(x):=\int \frac{f(y)}{|x-y|^{\alpha}}dy$ which majorizes $|Tf(x)|$ by the triangle inequality, when $f>0$. It asserts that $$ \|Hf\|_{q}\leq C \|f\|_{p}\label{1}\tag{1} $$ where $C$ is a constant independent of $f$, if and only if $$\frac{1}{q}-\frac{1}{p}=1-\alpha\label{2}\tag{2}$$ The necessity of \eqref{2} for \eqref{1} can be easily deduced by scaling.

My question:

Is \eqref{2} necessary for the inequality $$ \|Tf\|_{q}\leq C_{1} \|f\|_{p}\label{3}\tag{3}\;?$$

It seems that $T$ does not have any scaling properties. I cannot come up with counterexamples that show that \eqref{3} is false outside the range of exponents described by \eqref{2}. Is the boundedness of $T$ known ?