Pseudo-Differential and Fractional Schrödinger Operators on Grand Variable Herz–Morrey–Hardy Spaces with Muckenhoupt Weights

Fractional differential operators equipped with classical and new memory kernels have emerged as indispensable tools for modeling systems with hereditary and non-local effects, offering a flexible framework that extends far beyond the classical integer-order calculus. Within this broader landscape, pseudo-differential and Schrödinger operators play a very important role in mathematical analysis, arising naturally in the study of partial differential equations, quantum mechanics, and harmonic analysis, where they provide the fundamental tools for analyzing regularity, boundedness, and decay properties of solutions. The objective of this article is to define a new Herz norm with a variable exponent, a Morrey-type truncation, a grand modification, and an Ap(·) Muckenhoupt weight, and to prove two boundedness theorems. First, we show that every order-zero pseudo-differential operator is bounded on the resulting weighted grand variable Herz–Morrey–Hardy space HMK˙,p(·)α(·),υ)(w). Second, for the Schrödinger operator L=−+V with V in the reverse Hölder class RHq, q≥n/2, we replace the classical Riesz kernel with the Bessel–Riesz kernel and establish new, improved strong-type bounds for the associated Bessel–Riesz operator, from a weighted variable Lebesgue space into a weighted grand variable Herz–Morrey–Hardy–Lipschitz space built from the critical-radius geometry of L and using the same origin-centered dyadic scale as in the pseudo-differential case. Both proofs proceed via decompositions into uniform per-atom (respectively, per-ball) estimates, which are aggregated in the grand–Morrey norm under a decay condition on the Morrey parameter on the Herz–Hardy scale and under the threshold condition ≥1/υ on the Lipschitz-type range, together with a complementary condition on the Herz-type domain. We also show how the Bessel parameter γ enters the admissible range of integrability exponents. To further demonstrate the accuracy and consistency of these new results, we include remarks showing that, under different settings, various relevant results in the literature are recovered and improved, together with additional norm-structure and operator improvements.