The aim of this paper is to study the maximal commutators (M_{b}) and the commutators of the maximal operator [b, M] in the total Morrey spaces (L^{p,lambda ,mu }(mathbb {G})) on any stratified Lie group (mathbb {G}) when b belongs to Lipschitz spaces ({dot{Lambda }}_{beta }(mathbb {G})). Some new characterizations for certain subclasses of Lipschitz spaces ({dot{Lambda }}_{beta }(mathbb {G})) are given.
We develop a systematic approach to continuous substitutions on compact Hausdorff alphabets. Focussing on implications of irreducibility and primitivity, we highlight important features of the topological dynamics of their (generalised) subshifts. We then reframe questions from ergodic theory in terms of spectral properties of a corresponding substitution operator. This requires an extension of standard Perron–Frobenius theory to the setting of Banach lattices. As an application, we identify computable criteria that guarantee quasi-compactness of the substitution operator. This allows unique ergodicity to be verified for several classes of examples. For instance, it follows that every primitive and constant length substitution on an alphabet with an isolated point is uniquely ergodic, a result which fails when there are no isolated points.
In this paper we obtain necessary and sufficient conditions for the boundedness in weighted Lebesgue spaces of one-dimensional Hardy-type operators involving suprema. In particular, we solve the problems from Frank RL (J. Math. Sci. 263:323-342, 2022)
In this paper we study special properties of solutions of the initial value problem associated to a class of nonlinear dispersive equations where the operator modelling dispersive effects is nonlocal. In particular, we prove that solutions of the surface tension Whitham equations posed on the real line satisfy the propagation of regularity phenomena, which says that regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. A similar result is obtained for solutions of the Full Dispersion Kadomtsev–Petviashvili equation, a natural (weakly transverse) two-dimensional version of the Whitham equation, with and without surface tension. We establish that the augmented regularity of the initial data on certain distinguished subsets of the Euclidean space is transmitted by the flow solution at an infinite rate. The underlying approach involves treating the general equation as a perturbed version of a class of fractional equations with well-established properties.
We prove that p-harmonic systems with antisymmetric potentials of the form
((Omega ) is antisymmetric) can be written in divergence form as a conservation law
This extends to the p-harmonic framework the original work of the second author for (p=2) (see Rivière in Invent Math 168(1):1–22, 2007). We give applications of the existence of this divergence structure in the analysis (prightarrow 2).
In a previous paper, we proved a Carlson-type density theorem for zeroes in the critical strip for the Beurling zeta functions satisfying Axiom A of Knopfmacher. There we needed to invoke two additional conditions: the integrality of the norm (Condition B) and an “average Ramanujan condition” for the arithmetical function counting the number of different Beurling integers of the same norm (Condition G).
Here, we implement a new approach of Pintz using the classic zero-detecting sums coupled with Halász' method, but otherwise arguing in an elementary way avoiding, for example, large sieve-type inequalities or mean value estimates for Dirichlet polynomials. In this way, we give a new proof of a Carlson-type density estimate—with explicit constants—avoiding any use of the two additional conditions needed earlier.
Therefore, it is seen that the validity of a Carlson-type density estimate does not depend on any extra assumption—neither on the functional equation present for the Selberg class, nor on growth estimates of coefficients say of “average Ramanujan-type”—but is a general property presenting itself whenever the analytic continuation is guaranteed by Axiom A.
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