Mixed-Norm Amalgam Spaces and Their Predual

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

Singular integrals and sublinear operators on amalgam spaces and Hardy-amalgam spaces

In this paper, we establish the extrapolation theory for the amalgam spaces and the Hardy-amalgam spaces. By using the extrapolation theory, we obtain the mapping properties for the Calderón-Zygmund operators and its commutator, the Carleson operators and establish the Rubio de Francia inequalities for Littlewood-Paley functions of arbitrary intervals to the amalgam spaces. We also obtain the boundedness of the Calder{ó}n-Zygmund operators and the intrinsic square function on the Hardy-amalgam spaces.

Boundedness of Fractional Integral Operators on Hardy-Amalgam Spaces

We establish the boundedness of the fractional integral operators on the Hardy-amalgam spaces.

Martingale Transforms between Martingale Hardy-amalgam Spaces

We discuss martingale transforms between martingale Hardy-amalgam spaces H p , q s , Q p , q and P p , q . Let 0 < p < q < ∞ , p 1 < p and q 1 < q and let f = f n , n ∈ ℕ be a martingale in P p 1 , q 1 ; then, we show that its martingale transforms are the martingales in P p , q for some p , q and similarly for H p , q s and Q p , q .