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Taking the Fourier transform of the distributional equation givesbecause $2022$ is even. The multiplier therefore gives the regularity gain for one plus an even power of the Laplacian

The Sobolev embedding theorem in its derivative form says thatTo make all derivatives through order $4044$ continuous, and simultaneously make $f\in H^r$ continuous, it is enough and in general necessary to haveUnder this condition the distributional identity is a pointwise identity, so $u$ is a classical solution. ThusAt the borderline $n=2r$, the standard Sobolev embedding does not in general give continuity.