Symbol of a multiplier on \(L^{2}_{\omega}(\mathbb R)\) (Q707288)

It is considered the Hilbert space \(L^2_\omega (\mathbb R)\) of all (Lebesgue classes of) measurable functions \(f:\mathbb R \to \mathbb R\) on the real line \(\mathbb R\) such that \(\int_{\mathbb R} | f(x)| ^2 \omega (x)^2 \, dx < +\infty\), where \(\omega :\mathbb R \to \mathbb R\) is a weight, that is, \(\omega\) is a measurable, strictly positive function satisfying \(\widetilde \omega (t) :=\text{ess\,sup}_{x \in \mathbb R} {\omega (x+t) \over \omega (x)} < +\infty\) for all \(t \in \mathbb R\). A number of results concerning multipliers on Lebesgue spaces are known in the discrete case and in the case \(\omega = 1\). In the present paper, the author extends some of these to a general weight \(\omega\). Specifically, she defines \(M_\omega\) as the algebra of multipliers on \(L^2_\omega (\mathbb R)\), that is, \(M_\omega\) is the set of (continuous linear) operators \(T\) on \(L^2_\omega (\mathbb R)\) that commute with the translations \(S_{a, \omega}\), i.e., \(TS_{a, \omega} = S_{a, \omega}T\) \((a \in \mathbb R)\). For \(a \in \mathbb R\), the translation operator \(S_{a, \omega}\) on \(L^2_\omega (\mathbb R)\) is defined by \((S_{a, \omega} f)(x) = f(x-a)\). If \(a \in \mathbb R\) and \(f \in L^2_\omega (\mathbb R)\), then \((f)_a\) denotes the function \((f)_a: t \mapsto e^{at} f(t)\). As usual, \(D(\mathbb R)\) and \(L^\infty (\mathbb R)\) denote the space of real \(C^\infty\)-functions with compact support in \(\mathbb R\) and the space of (Lebesgue classes of) essentially bounded measurable real functions on \(\mathbb R\). In addition, it is denoted \(R^+_\omega := \lim_{x \to +\infty} \widetilde \omega (x)^{1/x} = \rho (S_{1,\omega})\) (\(\rho (A)\) is the spectrum of \(A\)), \(R^-_\omega := \lim_{x \to +\infty} \widetilde \omega (-x)^{-1/x} = 1/\rho (S_{-1,\omega})\), \(I_\omega := [\ln R^-_\omega ,\ln R^+_\omega ]\), \(\mathbb C :=\) the complex plane, \(G_\omega := \{z \in C: \ln R^-_\omega < \operatorname{Im} z <\) ln\(\, R^+_\omega\}\), \(H^\infty (G_\omega ) := \{\)bounded holomorphic functions on \(G_\omega\}\), \(\widehat f :=\) the Fourier transform of \(f\). The main result of the paper asserts that if \(T \in M_\omega\), then the following holds: (1) \((Tf)_a \in L^2(\mathbb R)\) for all \(f \in D(\mathbb R)\) and all \(a \in I_\omega\). (2) If \(a \in I_\omega\), there exists \(\nu_a \in L^\infty (\mathbb R)\) such that \(\widehat{(Tf)_a}(x) = \nu_a(x) \widehat{(f)_a}(x)\) almost everywhere, for every \(f \in D(\mathbb{R})\). In addition, there is a constant \(C_\omega \in (0,+\infty )\) depending only on \(\omega\) for which \(\| \nu_a\| _\infty \leq C_\omega \| T\| \) \((a \in \mathbb R)\). (3) If \(R^-_\omega < R^+_\omega\), there is \(\nu \in H^\infty (G_\omega )\) (\(\nu\) is the ``symbol'' of \(T\)) such that \(\widehat{Tf} = \nu \, \widehat f\) \((f \in D(\mathbb R))\), where \(\widehat{Tf} (a+ix) = \widehat{(Tf)_a}(x)\) for \(a \in I^0_\omega\), \(f \in D(\mathbb R)\).