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We prove that the sequence spaces $\ell_p\oplus\ell_q$ and the spaces ofinfinite matrices $\ell_p(\ell_q)$, $\ell_q(\ell_p)$ and$(\bigoplus_{n=1}^\infty \ell_p^n)_{\ell_q}$, which are isomorphic to certainBesov spaces, have an almost greedy basis whenever $0<p<1<q<\infty$. Moreprecisely, we custom-build almost greedy bases in such a way that the Lebesgueparameters grow in a prescribed manner. Our arguments critically depend on theextension of the Dilworth-Kalton-Kutzarova method from [S. J. Dilworth, N. J.Kalton, and D. Kutzarova, On the existence of almost greedy bases in Banachspaces, Studia Math. 159 (2003), no. 1, 67-101], which was originally designedfor constructing almost greedy bases in Banach spaces, to make it valid fordirect sums of mixed-normed spaces with nonlocally convex components.Additionally, we prove that the fundamental functions of all almost greedybases of these spaces grow as $(m^{1/q})_{m=1}^\infty$.

Existence of almost greedy bases in mixed-norm sequence and matrix spaces, including Besov spaces