In this work we generalize the spaces T u p introduced by Calderón and Zygmund using pointwise conditions emanating from generalized Besov spaces. We give conditions binding the functions belonging to these spaces and their wavelet coefficients. Next, we propose a multifractal formalism based on such spaces which generalizes the so-called wavelet leaders method and show that it is satisfied on a prevalent set.

In this paper, we present some alternative definitions of Besov spaces of generalized smoothness, defined via Littlewood–Paley‐type decomposition, involving weak derivatives, polynomials, convolutions and generalized interpolation spaces.

We study the Hölderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional Brownian Motion. Finally, we remark that the existence of slow points is specific to these functions.

We generalize the T p u spaces introduced by Calderón and Zygmund and show that most of the results obtained in their study of the pointwise estimates for solutions of elliptic partial differential equations and systems can be generalized in this framework with L p -conditions.

We consider the space $$C^1(K)$$ C 1 ( K ) of real-valued continuously differentiable functions on a compact set $$K\subseteq \mathbb {R}^d$$ K ⊆ R d . We characterize the completeness of this space and prove that the restriction space $$C^1(\mathbb {R}^d|K)=\{f|_K: f\in C^1(\mathbb {R}^d)\}$$ C 1 ( R d | K ) = { f | K : f ∈ C 1 ( R d ) } is always dense in $$C^1(K)$$ C 1 ( K ) . The space $$C^1(K)$$ C 1 ( K ) is then compared with other spaces of differentiable functions on compact sets.