On LpL^p--LqL^q trace inequalities

We give necessary and sufficient conditions in order that inequalities of the type $$\| T_K f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \qquad f \in L^p(d\sigma),$$ hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x, y) f(y) d \sigma(y)$ with nonnegative kernels, and measures $d \mu$ and $d\sigma$ on $\R^n$, in the case where $p>q>0$ and $p>1$. An important model is provided by the dyadic integral operator with kernel $K_{\mathcal D}(x, y) \sum_{Q\in{\mathcal D}} K(Q) \chi_Q(x) \chi_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\R^n$, and $K(Q)$ are arbitrary nonnegative constants associated with $Q \in{\mathcal D}$. The corresponding continuous versions are deduced from their dyadic counterparts. In particular, we show that, for the convolution operator $T_k f = k\star f$ with positive radially decreasing kernel $k(|x-y|)$, the trace inequality $$\| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d x)}, \qquad f \in L^p(dx),$$ holds if and only if ${\mathcal W}_{k}[\mu] \in L^s (d\mu)$, where $s = {\frac{q(p-1)}{p-q}}$. Here ${\mathcal W}_{k}[\mu]$ is a nonlinear Wolff potential defined by ${\mathcal W}_{k}[\mu](x)=\int_0^{+\infty} k(r) \bar{k}(r)^{\frac 1 {p-1}} \mu (B(x,r))^{\frac 1{p-1}} r^{n-1} dr,$ and $\bar{k}(r)=\frac1{r^n}\int_0^r k(t) t^{n-1} dt$. Analogous inequalities for $1\le q < p$ were characterized earlier by the authors using a different method which is not applicable when $q<1$

Small Hankel operators on generalized weighted Fock spaces

In this work we characterize the boundedness, compactness and membership in the Schatten class of small Hankel operators on generalized weighted Fock spaces $F^{p,\ell}_\alpha(\omega)$ associated to an $\mathcal{A}^\ell_p$ weight $\omega$, for $10$

Bilinear forms on non-homogeneous Sobolev spaces

In this paper we show that if $b\in L^2(\R^n)$, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces $H_s^2(\R^n)\times H_s^2(\R^n)$, $0<s<1$ by $$(f,g)\in H_s^2(\R^n)\times H_s^2(\R^n) \to \int_{\R^n} (Id-\Delta)^{s/2}(fg)({\bf x}) b({\bf x})d{\bf x},$$ is continuous if and only if the positive measure $|b({\bf x})|^2d{\bf x}$ is a trace measure for $H_s^2(\R^n)$

Bilinear forms on potential spaces in the unit circle

In this paper we characterize the boundedness on the product of Sobolev spaces $H^s(\mathbb{T}) \times H^s(\mathbb{T})$ on the unit circle $\mathbb{T}$, of the bilinear form $\Lambda_b$ with symbol $b \in H^s(\mathbb{T})$ given by $$ \Lambda_b(\varphi, \psi):=\int_{\mathbb{T}}\left((-\Delta)^s+I\right)(\varphi \psi)(\eta) b(\eta) d \sigma(\eta)$