Sharp Norm Phenomena and Duality in Hybrid Sobolev-Besov-Hardy Spaces: Non-Classical Operator Bounds and Applications to Rough PDEs

Abstract

This paper develops a unified theory of hybrid function
spaces that blend Sobolev, Besov and Hardy structures with
non-standard weights, addressing analytical gaps that arise
in problems with mixed regularity, anisotropy or rough
data. It establishes sharp embeddings with logarithmic
corrections, identifies new pathological duality behaviorsincluding asymmetric decompositions and non-measure
functionals-and reveals operatortheoretic phenomena
such as norm non-commutativity and phase transitions
tied to intersection order. Using new decomposition
methods combining wavelet analysis, paraproducts and
interpolation, the framework yields optimal regularity
results for degenerate elliptic operators, refined wellposedness thresholds for Navier-Stokes in critical hybrid
spaces and criteria for fractal singularity propagation
under fractional diffusion. It also resolves subtle stochastic
anomalies, notably explaining why Brownian convolutions
almost surely fail to belong to natural hybrid spaces despite
lying in each component space individually embedding theorems for functional spaces. Classical
theories of Sobolev spaces were later systematized by Adams
and Fournier [1], while Triebel [2] established the modern
framework for Besov and Hardy spaces. These foundations
were further developed by Peetre [11] in his innovative
work on Besov spaces, and Stein [12] in his comprehensive
treatment of Hardy spaces and singular integrals. The
Littlewood-Paley theory, as presented by Frazier, Jawerth
and Weiss [9], provided crucial decomposition techniques
that became indispensable for studying function spaces.
Subsequent advances by Runst and Sickel [8] demonstrated
important applications of these spaces to nonlinear partial
differential equations, particularly through their work
on Sobolev spaces of fractional order. Around the same
time, Lemarie-Rieusset [7] made significant contributions
to understanding function spaces in the context of fluid
dynamics, particularly for the Navier-Stokes equations.
The turn of the century saw major breakthroughs,
including Koch and Tataru’s [10] fundamental work on
well-posedness for Navier-Stokes equations in critical
spaces. Bourgain and Pavlovic [3] further advanced this
direction by studying ill-posedness in critical spaces.
Parallel developments in interpolation theory were made
by Hernandez and Yang [14], particularly regarding
Hardy-type spaces. Modern treatments by Maz’ya and
Shaposhnikova [17] on Sobolev multipliers and by
Bahouri, Chemin and Danchin [4] on nonlinear PDEs
have significantly expanded the applications of function