Littlewood--Paley theorem for arbitrary intervals: weighted estimates

S. V. Kislyakov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 180-198

Résumé

Suppose $1$ and $b$ is a weight on $\mathbb R$ such that $b^{-\frac1{r-1}}$ satisfies the Muckenhoupt condition $A_{r'/2}$ ($r'$ is the exponent conjugate to $r$). If $f_j$ are functions whose Fourier transforms are supported on mutually disjoint intervals, then $$ \Bigl\Vert\sum_j f_j\Bigr\Vert_{L^p(\mathbb R,b)}\le C\Bigl\Vert\Bigl(\sum_j|f_j|^2\Bigr)^{1/2}\Bigr\Vert_{L^p(\mathbb R,b)} $$ for $0$. Bibl. – 9 titles.