In this paper, we obtain non-testing characterizations, in terms of dyadic capacity gauges, of the boundedness and compactness of the differentiation operator $$ \frac{d}{dz}:Q_K\longrightarrow L^q(W\,dA), \qquad 0<q<\infty. $$ We also characterize the limiting case as $q\to0^+$, formulated in terms of a logarithmic geometric mean, while the endpoint $q=\infty$ is treated separately using a standard testing argument. These results greatly extend the previous work on ${\mathcal Q}_p$-spaces to the general setting of $Q_K$-spaces. As applications, we characterize composition operators and Volterra-type integral operators between different $Q_K$-spaces. In particular, the off-diagonal characterization established here, together with the previously established diagonal case, completely resolves Zhao's 2009 open question on composition operators between ${\mathcal Q}_p$-spaces.
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the M\"obius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mat...
UDC 517.54
In recent years, the $q$-difference operator has been primarily used to examine the characteristics of analytic functions. We~develop and analyze a unified $q$-analog of a family of linear operators arising in the geometric function theory. Our~primary objective is to generate some subclasses of analytic fun...
R. M. el-Ashwah, W. Y. Kota· Ukrains'kyi Matematychnyi Zh...· 0 citations
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
In this paper, we investigate the analytic properties of the Hardy-type operator $-\Delta + \mu H_0$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, where $\mu \geq -1$ and $H_0$ is a critical Hardy potential at infinity--arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green...
For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of th...
Let $X$ be a vector field on an open set of $\mathbb{R}^n$ with $X(p)=0$ and Jacobian $J=DX(p)$ of rank $n-1$ having $0$ as an eigenvalue of algebraic multiplicity two. Let $q_0$ span $\ker J$ and let $e_k(A)$ denote the sum of the principal $k\times k$ minors of $A$. Under the usual hyperbolicity assumption on the tra...
E. Chan-López· 1 citation
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