On the \(L^2\)-norm of periodizations of functions (Q2764646)

Let \(f\in L^1({\mathbb R}^d)\), \(d\geq 2\), and define its periodization with respect to the lattice \(\rho{\mathbb Z}^d\) (\(\rho \in SO(d)\), a rotation) to be the function NEWLINE\[NEWLINE g_\rho(x) = \sum_{\nu \in {\mathbb Z}^d} f(x-\nu). NEWLINE\]NEWLINE Define the mixed \(L^2\)-norm NEWLINE\[NEWLINE G^2 = \int_{{\mathbb R}^d} \int_{SO(d)} g_\rho(x) ~d\rho~dx. NEWLINE\]NEWLINE In this paper it is first shown how to bound the quantities \(||f||_2\) and \(G\) from each other and the quantity \(||f||_1\): (a) if \(d\geq 4\) one gets \(||f||_2 \leq C_d(G+||f||_1)\), (b) if \(d\geq 5\) one gets that \(G \leq C_d(||f||_2+||f||_1)\).NEWLINENEWLINENEWLINEThese results are then generalized for \(p\) in the range \(1 \leq p < 2d/(d+2)\) as follows: (c) if \(d\geq 4\) then \(||f||_2 \leq C_{d,p}(G+||f||_p)\), and, (d) if \(d\geq 5\) then NEWLINE\[NEWLINE \int_{SO(d)}||g_\rho-\widehat{g}(0)||_2^2~d\rho \leq C_{d,p}(||f||_2 + ||f||_p)^2. NEWLINE\]NEWLINE It is shown that in (d) one cannot remove the subtraction of the integral in the left hand side for \(p>1\).NEWLINENEWLINENEWLINESome further similar inequalities are also mentioned.