Abstract Let $$p(\cdot ):\Omega \rightarrow (0,\infty )$$ p ( · ) : Ω → ( 0 , ∞ ) be a variable exponent, $$0<q\le \infty $$ 0 < q ≤ ∞ and b be a slowly varying function. In this paper, we discuss the martingale theory of variable Lorentz-Karamata spaces $$L_{p(\cdot ), q,b}$$ L p ( · ) , q , b and apply it to Walsh-Fourier analysis. More precisely, we introduce the generalized BMO martingale spaces, which enable us to characterize the dual spaces of martingale Hardy spaces $$H^s_{p(\cdot ), q,b}$$ H p ( · ) , q , b s for $$0<p_-\le p_+<2$$ 0 < p - ≤ p + < 2 and $$0<q\le \infty $$ 0 < q ≤ ∞ . The John-Nirenberg theorem for the generalized BMO martingale spaces are presented by the dual results. We also investigate the boundedness of fractional integral operators on martingale Hardy spaces $$H^M_{p(\cdot ), q,b}$$ H p ( · ) , q , b M . As applications in Walsh-Fourier analysis, we consider the Walsh-Fourier series on variable Lorentz-Karamata spaces. The boundedness of maximal Fejér operator is proved, which further implies some convergence results of the Fejér means. The results obtained here generalize the known results in previous literature.

We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace transform. We start by applying this transform to the evolution problem, thus yielding a time-independent boundary value problem solely depending on the complex Laplace variable. In an offline stage, we judiciously sample the Laplace variable and numerically solve the corresponding collection of high-fidelity or full-order problems. Next, we apply a proper orthogonal decomposition (POD) to this collection of solutions in order to obtain a reduced basis in the Laplace domain. We project the linear parabolic problem onto this basis and then, using any suitable time-stepping method, we solve the evolution problem. A key insight to justify the implementation and analysis of the proposed method consists of using Hardy spaces of analytic functions and establishing, through the Paley-Wiener theorem, an isometry between the solution of the time-dependent problem and its Laplace transform. As a result, one may conclude that computing a POD with samples taken in the Laplace domain produces an exponentially accurate reduced basis for the time-dependent problem. Numerical experiments illustrate the performance of the method in terms of accuracy and, in particular, speed-up when compared to the solution obtained by solving the full-order model.

Block-Toeplitz Operators On the Hardy Space Induced by a Tracial Unital Banach $$*$$-Probability Space

Abstract Let $$\nu = (\nu _1, \ldots , \nu _n) \in (-1/2, \infty )^n$$ ν = ( ν 1 , … , ν n ) ∈ ( - 1 / 2 , ∞ ) n , with $$n \ge 1$$ n ≥ 1 , and let $$\Delta _\nu $$ Δ ν be the multivariate Bessel operator defined by $$ \Delta _{\nu } = -\sum _{j=1}^n\left( \frac{\partial ^2}{\partial x_j^2} - \frac{\nu _j^2 - 1/4}{x_j^2} \right) . $$ Δ ν = - ∑ j = 1 n ∂ 2 ∂ x j 2 - ν j 2 - 1 / 4 x j 2 . In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator $$\Delta _\nu $$ Δ ν . We then study the higher-order Riesz transforms associated with $$\Delta _\nu $$ Δ ν . First, we show that these transforms are Calderón-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with $$\Delta _\nu $$ Δ ν .

Let S H 0 ( K ) , K ≥ 1 , be the class of normalized K -quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses of S H 0 ( K ) . We then apply these results to obtain integral means estimates for the respective classes. Furthermore, we find the range of p > 0 such that these geometric classes of harmonic quasiconformal mappings are contained in the Hardy space h p , thereby refining some earlier results of Nowak.

Sharp weak-type estimate for local lifted Hardy–Littlewood maximal operators with applications to generators of linear operator families and Hardy(–Sobolev) spaces

The generalized continuous wavelet transform on BMO and Hardy spaces

In this paper we assume that $L = −∆_{\mathbb{H}^n}+ V$ is a Schrödinger operator on the Heisenberg group $\mathbb{H}^n,$ where the nonnegative potential $V$ belongs to the reverse Hölder class $B_{Q/2}.$ We introduce the Littlewood-Paley $\mathfrak{g}$-functions, the Lusin area functions and the $\mathfrak{g}^∗_λ$-functions generated by the heat semigroup $\{e^{−tL}\}_{t>0}$ and the Poisson semigroup $\{e^{−t\sqrt{L}$\}_{t>0},$ respectively. By means of the reproducing formulas and the regularity properties of semigroups, we establish several square function characterizations of the Hardy space $H_L^1(\mathbb{H}^n)$ associated with $L.$

Abstract Let X be a translation-invariant Banach function space on the unit circle $$\mathbb {T}$$ T with the associate space $$X'$$ X ′ , let w be a weight such that $$w\in X$$ w ∈ X and $$1/w\in X'$$ 1 / w ∈ X ′ , let X ( w ) consist of measurable functions $$f:\mathbb {T}\rightarrow \mathbb {C}$$ f : T → C such that $$fw\in X$$ f w ∈ X , and let H [ X ] and H [ X ( w )] denote the abstract Hardy spaces built upon X and X ( w ), respectively. Extending Rudin’s arguments (1962), we show that if $$\mathcal {P}$$ P is a bounded projection from X ( w ) onto H [ X ( w )], then the Riesz projection P is bounded from X onto H [ X ] and $$\Vert aI+bP\Vert _{\mathcal {B}(X)}\le \Vert aI+b\mathcal {P}\Vert _{\mathcal {B}(X(w))}$$ ‖ a I + b P ‖ B ( X ) ≤ ‖ a I + b P ‖ B ( X ( w ) ) for all $$a,b\in \mathbb {C}$$ a , b ∈ C . Further, for $$m\in \mathbb {N}$$ m ∈ N , let $$T(\textbf{e}_{-m})$$ T ( e - m ) be the Toeplitz operator with symbol $$\textbf{e}_{-m}(t)=t^{-m}$$ e - m ( t ) = t - m . We prove that $$\Vert T(\textbf{e}_{-m})\Vert _{\mathcal {B}(H[X])} \le \Vert T(\textbf{e}_{-m})\Vert _{\mathcal {B}(H[X(w)])}$$ ‖ T ( e - m ) ‖ B ( H [ X ] ) ≤ ‖ T ( e - m ) ‖ B ( H [ X ( w ) ] ) for all $$m\in \mathbb {N}$$ m ∈ N .

UDC 517.53 We explore Carleson measures given on the Hardy spaces defined over the bicomplex space and investigate the properties of the bicomplex Toeplitz operator. In the first section, we provide a comprehensive introduction and formulate the key definitions necessary for the subsequent analysis. In the second section, we present the results for $\mathbb{D}$-Carleson measures and establish their relationship with bicomplex Hardy spaces. Finally, in the last section, we extend the concept of the Toeplitz operator to the bicomplex space, thus offering broader prospects.

UDC 517.53, 517.58 The main objective of the present research is to examine a specific sufficiency criterion for the pre-starlikeness, strong starlikeness, strong convexity, starlikeness, and convexity of the generalized Bessel–Wright function. Furthermore, we determine the conditions for the generalized Bessel–Wright functions to be included in the Hardy spaces, as well as bounded analytic functions. These applications are given in the form of examples and corollaries.

Layer potential method for a Robin problem in Hardy spaces

Abstract Let be a Fourier integral operator of order associated with a canonical relation locally parametrised by a real‐phase function. A fundamental result due to Seeger, Sogge and Stein proved in the 90's gives the boundedness of from the Hardy space into . Additionally, it was shown by T. Tao the weak (1,1) type of . In this work, we establish the weak (1,1) boundedness of a Fourier integral operator of order when it has associated a canonical relation parametrised by a complex phase function. This result in the complex‐valued setting cannot be derived from its counterpart in the real‐valued case.

Area Operators on Anisotropic Mixed-Norm Hardy Spaces via Carleson Measures

We prove that singular integrals T with standard Calderón–Zygmund kernel and S with variable kernel are bounded on appropriate weighted Hardy spaces. Similar results hold for the commutators Tb and Sb when b belongs to a suitable subspace of BMO(Rn).

Calderón–Zygmund operators and commutators on weighted Hardy spaces

Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spaces

Abstract Let $$D\subset {\mathbb {C}}^n$$ D ⊂ C n be a bounded, strongly pseudoconvex domain whose boundary bD satisfies the minimal regularity condition of class $$C^2$$ C 2 . A 2017 result of Lanzani & Stein [17] states that the Cauchy–Szegő projection $$\mathcal {S}_\omega $$ S ω defined with respect to a bounded, positive continuous multiple $$\omega $$ ω of induced Lebesgue measure, maps $$L^p(bD, \omega )$$ L p ( b D , ω ) to $$L^p(bD, \omega )$$ L p ( b D , ω ) continuously for any $$1<p<\infty $$ 1 < p < ∞ . Here we show that $$\mathcal {S}_\omega $$ S ω satisfies explicit quantitative bounds in $$L^p(bD, \Omega _p)$$ L p ( b D , Ω p ) , for any $$1<p<\infty $$ 1 < p < ∞ and for any $$\Omega _p$$ Ω p in the maximal class of $$A_p$$ A p -measures, that is for $$\Omega _p = \psi _p\sigma $$ Ω p = ψ p σ where $$\psi _p$$ ψ p is a Muckenhoupt $$A_p$$ A p -weight and $$\sigma $$ σ is the induced Lebesgue measure (with $$\omega $$ ω ’s as above being a sub-class). Earlier results rely upon an asymptotic expansion and subsequent pointwise estimates of the Cauchy–Szegő kernel, but these are unavailable in our setting of minimal regularity of bD ; at the same time, more recent techniques that allow to handle domains with minimal regularity [17] are not applicable to $$A_p$$ A p -measures. It turns out that the method of quantitative extrapolation is an appropriate replacement for the missing tools. To finish, we identify a class of holomorphic Hardy spaces defined with respect to $$A_p$$ A p -measures for which a meaningful notion of Cauchy-Szegő projection can be defined when $$p=2$$ p = 2 .

Composition operators between Beurling subspaces of Hardy space

Fractional Volterra-type operator induced by radial weight acting on Hardy space