Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces

In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that $$M_0=0$$ M 0 = 0 , we show that the following two-sided inequality holds for all $$1\le p<\infty $$ 1 ≤ p < ∞ : Here $$ \gamma ([\![M]\!]_t) $$ γ ( [ [ M ] ] t ) is the $$L^2$$ L 2 -norm of the unique Gaussian measure on X having $$[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle , \langle M,y^*\rangle ]_t$$ [ [ M ] ] t ( x ∗ , y ∗ ) : = [ ⟨ M , x ∗ ⟩ , ⟨ M , y ∗ ⟩ ] t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of ($$\star $$ ⋆ ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, ($$\star $$ ⋆ ) holds for all $$0<p<\infty $$ 0 < p < ∞ , and that for purely discontinuous martingales the right-hand side of ($$\star $$ ⋆ ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, ($$\star $$ ⋆ ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ($$\star $$ ⋆ ) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

Banach-Valued Multilinear Singular Integrals with Modulation Invariance

We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $\operatorname{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\operatorname{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\textrm{UMD}$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $\textrm{UMD}$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors.

The bilinear Hilbert transform in UMD spaces

We prove $$L^p$$ L p -bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $$L^p(\mathbb {R};X)$$ L p ( R ; X ) into outer Lebesgue spaces on the time-frequency-scale space $$\mathbb {R}^3_+$$ R + 3 .

Weighted estimates for operator-valued Fourier multipliers

We establish Littlewood–Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions.