Generalized multiplicative Hilbert operators on Hilbert spaces of Dirichlet series

This paper investigates the generalized multiplicative Hilbert operator $\mathbf H_g$ defined by $$\mathbf H_g f(s)=\int _0^\infty f(1/2+\sigma )g’(s+\sigma )\,d\sigma $$ in weighted Hilbert spaces of Dirichlet series $\mathcal A_{\alpha }^2$. Specifically, the multiplicative Hilbert matrix corresponds to $\mathbf H_{\mathbf g}$ with $\mathbf g(s)=-\sum _{n=2}^\infty \frac {n^{-s}}{\sqrt {n}\log n}$. To characterize the boundedness of $\mathbf H_g$, we introduce the mean Lipschitz space of Dirichlet series $\Lambda (p,\gamma )$ and provide characterizations of functions in $\Lambda (p,\gamma )$ in terms of horizontal translations, derivatives and logarithmic dyadic sums. Based on these characterizations and the local embedding theorem for $\mathcal A_{\alpha }^2$, we demonstrate that for $1/2 \lt \alpha \lt 2$ the operator $\mathbf H_g$ is bounded on $\mathcal A_{\alpha }^2$ if and only if $g$ belongs to $\Lambda (2,1/2)$.