Inclusion Criteria:
 - -  Participants must be able to understand and willing to sign a written informed
     consent document.
- -  Participant must have signed and dated an IRB/IEC approved written informed consent
     form in accordance with regulatory and institutional guidelines.
This must be
     obtained before the performance of any protocol-related procedures that are not part
     of normal subject care.
 
- -  Participant must be willing and able to comply with scheduled visits, treatment
     schedule, laboratory tests, and other requirements of the study.
- -  Participants must be at least 18 years old on day of signing informed consent.
- -  Participants must have a Karnofsky Performance Status (KPS) ≥ 70.
- -  Participants must have histologically confirmed R/R primary DLBCL CNS lymphoma (from
     brain biopsy, CSF or vitreous biopsy).
- -  Participants should have evidence of refractory or recurrent disease on MRI with
     measurable or evaluable enhancing disease.
- -  Participants must have recovered to ≤ grade 1 or pre-treatment baseline from
     clinically significant toxic effects of prior therapy; exception, participants with
     ≤ grade 2 neuropathy may be eligible.
- -  Participant with dexamethasone requirement of ≤ 8mg/day or bioequivalent with
     corticosteroid usage at a stable or decreasing dose 2 weeks prior to screening.
- -  Participants must be able to undergo MRI.
- -  Participants must demonstrate adequate as defined below (all screening labs should
     be performed within 14 days of treatment initiation):
       -  Hematology.
- -  White Blood Count (WBC) ≥ 2 K/µL.
- -  Platelet count ≥ 100 K/µL.
- -  Absolute Neutrophil Count ≥ 1.5 K/µL.
- -  Hemoglobin > 9.0 g/dL or ≥ 5.6 mmol/L (Criteria must be met without
               erythropoietin dependency and without packed red blood cell (pRBC)
               transfusion within last 2 weeks)
       -  Biochemistry.
- -  Serum creatinine ≤1.5 x institutional ULN OR Measured or calculated
               creatinine clearance ≥30 mL/min for participant with creatinine levels
               >1.5 × institutional ULN (Creatinine clearance should be calculated per
               institutional standard)
            -  Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 2.5
               x ULN (≤5 × ULN for participants with liver metastases)
            -  Total bilirubin (TBILI) ≤ 1.5 x institutional ULN (except subjects with
               Gilbert Syndrome who must have a total bilirubin level of < 3.0 x
               institutional ULN) OR Direct bilirubin ≤ULN for participants with total
               bilirubin levels >1.5 × ULN)
  -  Women of childbearing potential (WOCBP) must have a negative serum pregnancy within
     72 hours prior to registration.
- -  WOCBP who are sexually active must use highly effective methods of contraception
     during treatment and for 28 days after the last dose of paxalisib.
For male subjects
     with a pregnant or non-pregnant WOCBP partner, contraception measures are required
     during treatment and for 28 days after the last dose of paxalisib.
The subject, in consultation with the investigator, will select the most appropriate
method of contraception from the permitted list of contraception methods, and site
personnel will instruct the subject in its consistent and correct use as needed.
In addition, the investigator will instruct the subject to notify the site immediately if
pregnancy of the subject or their partner is known or suspected.
Highly effective methods of contraception are those that, alone or in combination, result
in a failure rate of less than 1% per year when used consistently and correctly and
include:
 
- -  Established use of oral, injected, or implanted hormonal methods of contraception.
- -  Correctly placed intrauterine device (IUD) or intrauterine system (IUS)
  -  Male condom or female condom used WITH a spermicide (i.e., foam, gel, film, cream)
  -  Male sterilization with appropriately confirmed absence of sperm in the
     post-vasectomy ejaculate.
- -  Bilateral tubal ligation or bilateral salpingectomy.
Exclusion Criteria:
 - -  Participants unable to undergo MRI brain.
- -  Participants with active systemic disease.
- -  Participants with uncontrolled intercurrent illness.
- -  Participants with prior exposure to mTOR/PI3K inhibitors.
- -  Prior malignancy (or any other malignancy requiring active treatment), except for
     adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer,
     superficial bladder cancer or other cancer from which the subject has been disease
     free for ≥ 3 years.
- -  Participants who have received prior systemic anti-cancer therapy including
     investigational agents or radiotherapy within 4 weeks OR 5 half-lives prior to
     dosing, whichever is shorter.
Note: Participants must have recovered from all AEs
     due to previous therapies to ≤Grade 1 or baseline. Participants with ≤Grade 2
     neuropathy may be eligible.
 
- -  Participants who have difficulty with or are unable to swallow oral medication or
     have significant gastrointestinal disease that would limit absorption of oral
     medication.
- -  Known history of infection with HIV, prior history of PML or any active significant
     infection (eg, bacterial, viral, or fungal).
- -  Known history of hypersensitivity or anaphylaxis to paxalisib including active
     product or excipient components.
- -  Requires treatment with a strong cytochrome P450 3A4 (CYP3A4) inhibitor/inducer
     which may have an effect of the metabolism of paxalisib.
- -  Participants with uncontrolled medical comorbidities per investigator discretion
     including but not limited to interstitial lung disease, severe dyspnea at rest or
     requiring oxygen therapy, pre-exisiting Crohn's disease or ulcerative colitis or
     pre-existing chronic condition resulting in baseline grade 2 or higher diarrhea.
- -  Participants with type I diabetes mellitus, participants with uncontrolled type II
     diabetes mellitus,despite being on oral anti-diabetic medication.
, participants
     with Type II diabetes mellitus that are well controlled on insulin . Uncontrolled
     diabetes is defined as HbA1c >9% in addition to fasting glucose>140mg/dL on at least
     2 occasions within 14 days prior to registration. 
- -  Participants with uncontrolled hypertension despite optimal medical management (per
     investigator's assessment).
- -  Hepatitis B or C serologic status: subjects who are hepatitis B core antibody
     (anti-HBc) positive and who are hepatitis B surface antigen (HBsAg) negative will
     need to have a negative polymerase chain reaction (PCR) and must be willing to
     undergo DNA PCR testing during the study to be eligible.
Those who are HBsAg
     positive or hepatitis B PCR positive will be excluded. Subjects who are hepatitis C
     antibody positive will need to have a negative PCR result to be eligible. Those who
     are hepatitis C PCR positive will be excluded.
 
- -  Breast feeding or pregnant.
- -  Concurrent participation in another therapeutic trial.