Some convolution inequalities and their applications (Q2781418)

Let \(\lambda\) be a nonnegative Borel measure. Two problems are of great importance in harmonic analysis. 1) Determine the indices \(p\) and \(q\) such that the convolution estimate NEWLINE\[NEWLINE\|\lambda\ast f\|_q\leq C(\lambda,p,q)\|f\|_pNEWLINE\]NEWLINE holds for \(f\in L^p(\mathbb R^n).\) 2) Determine the indices \(p\) and \(q\) such that the adjoint Fourier restriction estimate NEWLINE\[NEWLINE\|\widehat{fd\lambda}\|_q\leq C(\lambda,p,q)\|f\|_{L^p(\lambda)}NEWLINE\]NEWLINE holds for \(f\in L^p(\mathbb R^n).\) In the paper under review, these problems are studied for degenerated, in a sense, cases. Among 7 theorems proven in the paper we here give two very general.NEWLINENEWLINENEWLINE\textbf{Theorem 5.} Suppose that \(\lambda\) is a nonnegative Borel measure on \(\mathbb R^n\) satisfying the inequality NEWLINE\[NEWLINE\|\lambda\ast\chi_E\|_{L^m(\lambda)} \leq c|E|^{1-1/m}NEWLINE\]NEWLINE for \(E\subseteq{\mathbb R^n}.\) Then the convolution estimate NEWLINE\[NEWLINE\|\lambda\ast f\|_{m+1}\leq C(c)\|f\|_{1+1/m}NEWLINE\]NEWLINE holds for \(f\in L^{1+1/m} (\mathbb R^n).\) NEWLINENEWLINENEWLINE\textbf{Theorem 6.} Suppose that \(\lambda\) is a nonnegative Borel measure on \(\mathbb R^n.\) Suppose that \(\omega\) is a real-valued Borel function on supp( \(\lambda\)) and suppose the inequality NEWLINE\[NEWLINE\int\left(\int\limits_{\{\omega(y_1) \geq\omega(y_2)\}}\chi_E(y_2-y_1)d\lambda(y_1)\right)^m d\lambda(y_2)\leq c|E|^{m-1}NEWLINE\]NEWLINE holds for \(E\subseteq{\mathbb R^n}.\) Then the convolution estimate NEWLINE\[NEWLINE\|\lambda\ast\chi_E\|_{m+1}\leq C(c)\|f\|_{m\over m+1}NEWLINE\]NEWLINE holds whenever \(E\subseteq{\mathbb R^n}.\) NEWLINENEWLINENEWLINEOne of the features of this interesting paper is the numerous examples.