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Let X be a rearrangement-invariant space of Lebesgue-measurable functions on ${ℝ}^n$, such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on ${ℝ}^n$, define $X\left(w\right)=F:{ℝ}^n\to ℂ:\infty >\parallel F{\parallel }_{X\left(w\right)}:=\parallel Fw{\parallel }_X$. We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at $x\in {ℝ}^n$ by $\left(F\ast G\right)\left(x\right)={ʃ}_{{ℝ}^n}F(x-y)G\left(y\right)dy$; more precisely, when $\parallel F\ast G{\parallel }_{X\left(w\right)}\le \parallel F{\parallel }_{X\left(w\right)}\parallel G{\parallel }_{X\left(w\right)}$ for all F,G ∈ X(w).

Necessary and sufficient conditions are given for the Hardy-Littlewood maximal operator to be bounded on a weighted Orlicz space when the complementary Young function satisfies ${\Delta }_2$. Such a growth condition is shown to be necessary for any weighted integral inequality to occur. Weak-type conditions are also investigated.

Suppose $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^n$, which is nonincreasing in $\left|x\right|$. Set $\left(Tf\right)\left(x\right)={\int }_{R^n}\Phi (x-y)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^n$. Given $1\<p\<\infty$ and $v\ge 0$, we show there exists $C\>0$ so that ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ for all $f\ge 0$, if and only if $C^{\text{'}}\>0$ exists with ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\text{'}}}dx\le C^{\text{'}}{\int }_{Q}v\left(x\right)dx\<\infty$ for all dyadic cubes Q, where $p^{\text{'}}=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of ⨍, $M_{\gamma }⨍$, by an expression involving the nonincreasing rearrangement of ⨍. This estimate is used to obtain necessary and sufficient conditions for the boundedness of $M_{\gamma }$ between classical Lorentz spaces.